Bilinear Form Research Articles

Let $Q$ be the field of rational numbers and $Q^{2}$ be the $2$-dimensional linear space over $Q$. A classification of all non-degenerate symmetric bilinear-metric forms over $Q^{2}$ have obtained. Let $\varphi $ be a non-degenerate symmetric bilinear form on $Q^{2}$. Denote by $O(2,\varphi, Q)$ the group of all $\varphi$-orthogonal (that is the form $\varphi$ preserving) transformations of $Q^{2}$. Put $MO( 2,\varphi, Q)=\left\{F:Q^{2}\rightarrow Q^{2}\mid Fx=gx+b, g\in O(2,\varphi, Q), b\in Q^{2}\right\}$, $SO(2,\varphi, Q)=\left\{ g\in O(2, \varphi, Q)|detg=1\right\}$ and $MSO(2, \varphi, Q)= \left\{F\in M(2, \varphi, Q)|detg=1\right\}$.The present paper is devoted to solutions of problems of $G$-equivalence of $m$-tuples in $Q^{2}$ for groups $G=O(2,\varphi, Q), SO(2,\varphi, Q)$, $MO(2,\varphi, Q)$, $MSO(2,\varphi, Q)$. Complete systems of $G$-invariants of $m$-tuples in $Q^{2}$ for these groups are obtained.

In this paper, we first classify all associative algebra structures on a two-dimensional vector space over an arbitrary base field equipped with a non-degenerate bilinear form. We then determine which of these are Frobenius algebras. We provide lists of canonical representatives of the isomorphism classes of these algebras over an arbitrary base field.

In this study, we consider a (2+1)-dimensional integrable Boussinesq equation, where the Hirota method of positive logarithmic transformation is used to convert it into a bilinear form. We proceeded by employing different test functions, through which we obtained breather solutions, two-wave solutions, lump-periodic solutions, and new interaction solutions. The resulting soliton dynamics for the governing model are also derived using the enhanced modified extended tanh function method, where varieties of solutions, such as trigonometric, hyperbolic, and rational forms, were obtained. The derived solutions may hold significant potential for explaining real-world physical phenomena in fields like mathematical physics, plasma physics, and nonlinear optics. The accuracy and reliability of the solutions were tested by substituting them back into the original equation using Python, highlighting the method’s robustness, precision, and reliability. By choosing appropriate physical parameters, we showcased the rich diversity and dynamic behavior of the obtained soliton structures. In other words, the graphical representations in 3D, contour, and 2D were provided for some of the obtained results. The modulation instability analysis and gain spectrum of the model are also provided. The importance of the obtained results in the area of (2+1)-dimensional integrable equation application was also highlighted.

In this study, we construct soliton solutions to the space-time fractional (3 + 1)-dimensional Jimbo-Miwa (JM) equation with a modified Riemann-Liouville derivative. Analysing this equation holds significant importance due to its utility in clarifying nonlinear wave behaviours in a variety of physical settings, such as plasma science, nuclear explosions, the solar system, tsunami waves and lightning pressure blasts. Using the Hirota bilinear method in conjunction with the homoclinic test technique and a complex transformation, based on the modified Riemann–Liouville fractional derivative, exact soliton solutions for the given equation are successfully established. A simplified algorithm is proposed to convert any nonlinear fractional partial differential equation into Hirota's bilinear form through complex transformation and the Cole-Hopf transformation. First, we apply the fractional complex transformation to the aforementioned equation to convert it into a solvable classical ordinary differential equation. After successfully implementing the proposed algorithm, multi-soliton, multi-kink, lump, breather and rough wave solutions are obtained for the fractional JM equation. To further explore the dynamical characteristics of the model, complex hybrid solutions, such as kink-breather and kink-lump wave solutions, are investigated. Furthermore, the impact of fractional parameters, along with other variables, on various wave structures is analysed using numerical graphs. The study illustrates different dynamic behaviours of solitons and their collisions as the fractional parameters change.

This paper studies discontinuous quasilinear sub-elliptic systems associated with Hörmander’s vector fields under controllable and natural growth conditions. By a new A-harmonic approximation reformulation for bilinear forms A∈Bil(RkN,RkN), we obtain optimal partial Hölder continuity with exact exponents for weak solutions with vanishing mean oscillation coefficients.

Abstract We study local algebras, which are structures similar to $$\mathbb {Z}$$ Z -graded algebras concentrated in degrees $$-1,0,1$$ - 1 , 0 , 1 , but without a product defined for pairs of elements at the same degree $$\pm 1$$ ± 1 . To any triple consisting of a Kac–Moody algebra $${\mathfrak g}$$ g with an invertible and symmetrisable Cartan matrix, a dominant integral weight of $${\mathfrak g}$$ g and an invariant symmetric bilinear form on $${\mathfrak g}$$ g , we associate a local algebra satisfying a restricted version of associativity. From it, we derive a local Lie superalgebra by a commutator construction. Under certain conditions, we identify generators which we show satisfy the relations of the tensor hierarchy algebra W previously defined from the same data. The result suggests that an underlying structure satisfying such a restricted associativity may be useful in applications of tensor hierarchy algebras to extended geometry.

This paper considers a linear system of partial differential equations (PDEs) to describe the stress-strain state of a two-phase body under static load, such as water-saturated soil. It investigates the basic properties of a new general differential operator Lame. The equations differ from the classical Lame equations by including first derivatives, which account for the influence of pore water on soil mineral particles. The properties of the generalized Lamé operator are investigated for the application of variational methods to solve the problem. It also describes alternative of the Betti and Clapeyron formulas using strain energy results. The calculus of variations of the Galerkin method is used to solve the minimum functional problem. Properties of bilinear forms are established and a theorem on the existence and uniqueness of the solution of the two-phase equilibrium problem is proved. The finite element method is adapted for a kinematic model that considers excess residual pore pressures. A new stiffness matrix is obtained, which is the sum of two matrices: one for the soil skeleton and one for pore water. The adequacy of the mathematical model of a water-saturated foundation for a natural experiment is shown. The use of Korn's inequality implies limitations on elastic properties (homogeneity, anisotropy) and the geometry of the region (requiring regularity and smooth boundaries). The study illustrates that the methodology of mechanics of a deformable solid can be adapted with appropriate modifications to a two-phase body in a stabilized state. The finite element method is adapted for a kinematic model that considers excess residual pore pressures. A new stiffness matrix is obtained, which is the sum of two matrices: one for the soil skeleton and one for pore water. The finite element method is tested on the Flamand problem. The adequacy of the mathematical model of a water-saturated foundation for a full-scale experiment is shown. The problem of the action of distributed load on a water-saturated heterogeneous foundation was solved using the finite element method and the results were compared with experimental data. The effect of mesh partitioning on the accuracy of the numerical solution is also studied in the finite element method. The maximum discrepancy was no more than 26%.

Abstract This paper focuses on quadratic Hom–Leibniz algebras, defined as (left or right) Hom–Leibniz algebras equipped with symmetric, non-degenerate, and invariant bilinear forms. In particular, we demonstrate that every quadratic regular Hom–Leibniz algebra is symmetric, meaning that it is simultaneously a left and a right Hom–Leibniz algebra. We provide characterizations of symmetric (resp. quadratic) Hom–Leibniz algebras. We also investigate the $\mathrm{T}^*$ -extensions of Hom–Leibniz algebras, establishing their compatibility with solvability and nilpotency. We study the equivalence of such extensions and provide the necessary and sufficient conditions for a nilpotent quadratic Hom–Leibniz algebra to be isometric to a $\mathrm{T}^*$ -extension. Furthermore, through the procedure of double extension, which is a central extension followed by a generalized semi-direct product, we get an inductive description of all quadratic regular Hom–Leibniz algebras, allowing us to reduce their study to that of quadratic regular Hom–Lie algebras. Finally, we construct several non-trivial examples of symmetric (resp. quadratic) Hom–Leibniz algebras.

Massless chiral fields of arbitrary spin in six spacetime dimensions, also known as higher spin singletons, admit a simple formulation in terms of SU^*(4) \cong SL(2,\mathbb{H})SU*(4)≅SL(2,ℍ) tensors. We show that, paralleling the four-dimensional case, these fields can be described using a 00-form and a gauge 22-form, taking values in totally symmetric tensors of SU^*(4)SU*(4). We then exhibit an example of interacting theory that couples a tower of singletons of all integer spin to a background of \mathfrak{g}𝔤-valued higher spin fields, for \mathfrak{g}𝔤 an arbitrary Lie algebra equipped with an invariant symmetric bilinear form. Finally, we discuss the formulation of these models in arbitrary even dimensions, as well as their partially-massless counterpart.

Purpose This paper aims to investigate the (3 + 1)-dimensional integrable extension Hirota bilinear (3D-eHB) equation, which models nonlinear wave phenomena in oceans. Design/methodology/approach This study integrates the Hirota bilinear form with various test functions to construct lump-multi-stripe and lump-multi-soliton solutions for the 3D-eHB equation. Additionally, it formulates interaction solutions that are characterized by the interplay between lump, periodic and kinky waves. Findings The fusion and fission of lump-multi-stripe and lump-multi-soliton solutions are studied. Additionally, the dynamic features of three distinct categories of interaction solutions are analyzed through numerical simulations. Originality/value This paper constructs novel exact solutions for the 3D-eHB equation using various test functions. It explores the interaction phenomena between one lump and multi-stripe waves, between one lump and multi-soliton waves and among lump, periodic and kinky waves.

Abstract Employing the gauge-invariant formalism in the two-Higgs-doublet model (THDM) offers profound insights into the model’s fundamental structure. A specific set of gauge-invariant bilinear combinations, constructed from the Higgs doublets, establishes a one-to-one correspondence between the components of the doublet fields and real-valued bilinears. This formalism provides a compact and consistent framework to study various aspects of the THDM, including stability, electroweak symmetry breaking, basis transformations, and general symmetries of the Higgs potential. Recently, the bilinear formalism has been extended beyond the Higgs potential to encompass the full THDM, including the gauge and Yukawa sectors, all in gauge-invariant terms. In this work, we advance the formalism further by incorporating quantum corrections. Specifically, we show how bilinears, combined with the ħ-expansion, can be used to compute one-loop corrections. We provide concise, gauge-invariant expressions for these corrections, which are directly applicable to the THDM.

It is known that there are two inequivalent [Formula: see text]-graded [Formula: see text] Lie superalgebras. Their affine extensions are investigated and it is shown that one of them admits two central elements, one is non-graded and the other is [Formula: see text]-graded. The affine [Formula: see text]-[Formula: see text] algebras are used by the Sugawara construction to study possible [Formula: see text]-graded extensions of the Virasoro algebra. We obtain a [Formula: see text]-graded Virasoro algebra with a non-trivially graded central element. Throughout the investigation, invariant bilinear forms on [Formula: see text]-graded superalgebras play a crucial role, so a theory of invariant bilinear forms is also developed.

We compute the Brauer group of surfaces defined by equating two bilinear forms of the same degree, assuming these forms are, in an explicit sense, sufficiently general. Our method uses a topological deformation argument and does not require full knowledge of the algebraic or transcendental cycles. We obtain a criterion for the triviality of the transcendental Brauer group of an isotrivial variety, which we use to prove that the Brauer group of the generic diagonal surface of arbitrary degree is trivial.

Abstract We study the half-sided translations associated to Rindler wedge algebras for conformal field theories in 1+1 Minkowski spacetime, generated by an unbounded operator $$ \mathcal{G} $$ G , in terms of bilinear forms G, G′ made from entanglement Hamiltonians of the underlying algebras such that $$ \mathcal{G} $$ G = G + G′. We show that despite entanglement Hamiltonians being ill-defined operators on Hilbert space, G, G′ can be regularized using smooth bump functions to operators $$ \hat{G} $$ G ̂ , $$ {\hat{G}}^{\prime } $$ G ̂ ′ with well-defined commutators, and use them to do a centered Zassenhaus expansion of exp ( $$ i\mathcal{G}s $$ i G s ) in terms of $$ \hat{G} $$ G ̂ and $$ {\hat{G}}^{\prime } $$ G ̂ ′ which is tractable and respects causality. We show that in fact half-sided translations is a special case in a large class of operators $$ \mathcal{O} $$ O for which a similar decomposition can be done by defining $$ \mathcal{O} $$ O = O L + O R with O L , O R chosen approriately.

Many studies have aimed to improve Transformer model efficiency using low-rank-based methods that compress sequence length with predetermined or learned compression matrices. However, these methods fix compression coefficients for tokens in the same position during inference, ignoring sequence-specific variations. They also overlook the impact of hidden state dimensions on efficiency gains. To address these limitations, we propose dynamic bilinear low-rank attention (DBA), an efficient and effective attention mechanism that compresses sequence length using input-sensitive dynamic compression matrices. DBA achieves linear time and space complexity by jointly optimizing sequence length and hidden state dimension while maintaining state-of-the-art performance. Specifically, we demonstrate through experiments and the properties of low-rank matrices that sequence length can be compressed with compression coefficients dynamically determined by the input sequence. In addition, we illustrate that the hidden state dimension can be approximated by extending the Johnson-Lindenstrauss lemma, thereby introducing only a small amount of error. DBA optimizes the attention mechanism through bilinear forms that consider both the sequence length and hidden state dimension. Moreover, the theoretical analysis substantiates that DBA excels at capturing high-order relationships in cross-attention problems. Experimental results across different tasks with varied sequence length conditions demonstrate that DBA achieves state-of-the-art performance compared to several robust baselines. DBA also maintains higher processing speed and lower memory usage, highlighting its efficiency and effectiveness across diverse applications.

The dust acoustic waves with (r, q) distributed ions and Maxwellian electrons are studied for dusty plasmas found in planetary rings. The expression of polarization force in a dusty plasma, modified due to (r, q) distributed ions, is evaluated for the first time. The fundamental model equations are reduced to a modified Korteweg–de Vries (mKdV) equation by applying the familiar reductive perturbation technique. The dynamical system analysis provides phase portraits, effective potential, compressive and rarefactive solitary waves, and nonlinear periodic waves for the dusty plasma. Hirota's bilinear formalism is utilized to generate the multi-soliton solutions. The parametric analysis of dust acoustic solitary waves (DASWs) shows that the modified polarization force tends to augment the amplitude of the solitary waves. The comparison of kappa, (r, q) and Maxwellian distributions shows that the DASWs with kappa distribution exhibit minimum amplitude, while DASWs with higher values of indices in (r, q) ion distribution manifest maximum amplitude of solitary waves. The Hirota breather is formed using the complex-conjugate pair of propagation vectors for the two-soliton solution of the mKdV equation. The interaction between solitons, between a breather and a soliton, and between two breathers shows that such interactions have nonlinear superposition characteristics. This shows that the breather solutions are stable solutions, like solitons, for an integrable nonlinear partial differential equation (nPDE). The model is applied to the dusty space plasmas; however, the soliton and breather solutions obtained in this paper are general and may be applied to the systems where integrable nPDEs are obtained.

Abstract In this article, we delve into the variable coefficient generalized coupled nonlinear Schrödinger (vcGCNLS) equations with four-wave mixing terms in the context of a birefringent fiber, encompassing time-dependent diffraction, nonlinearity, and gain (loss) parameters. Utilizing the Hirota bilinear method, we derive the bilinear forms of the equations. Subsequently, we calculate the nondegenerate one-soliton and two-solitons solutions. Thereafter, by adjusting the complex wave parameters, we analyze the dynamic behaviors of solitons. Additionally, through varying the diffraction parameter and nonlinear coefficient, we successfully obtain nondegenerate double-hump solitons of diverse shapes.

This study investigates the complex dynamics of the time-fractional Phi-four model, a nonlinear PDE that incorporates memory effects through fractional derivatives. To analyze this model, we employ the Hirota bilinear method to derive a variety of exact analytical solutions, including one-wave, two-wave, three-wave, and W-shaped soliton solutions, as well as lump-type solutions such as lumps, one-lump-one-stripe, and one-lump-one-soliton configurations. We conduct a detailed analysis of the system’s chaotic behaviour by calculating Lyapunov exponents, performing sensitivity analysis, and constructing bifurcation diagrams, which reveal transitions between stable, periodic, and chaotic states. The results demonstrate that the fractional-order derivative crucially influences system dynamics by introducing memory effects that can stabilize or destabilize wave propagation. A linear stability analysis confirms the conditions under which these soliton and lump solutions remain structurally stable against perturbations. These findings advance the understanding of nonlinear fractional systems by illustrating their capacity for rich wave interactions and chaotic dynamics.

In the present work, we give a new proof of the well-known Selberg’s inequality in complex Hilbert spaces from an operator-theoretic perspective, establishing its fundamental equivalence with the Cauchy–Bunyakovsky–Schwarz inequality. We also derive several lower and upper bounds for the Selberg operator, including its norm estimates, refining classical results such as de Bruijn’s and Bohr’s inequalities. Additionally, we revisit a recent claim in the literature, providing a clarification of the conditions under which Selberg’s inequality extends to abstract bilinear forms.

Abstract This paper investigates the localized waves of the Hirota-Satsuma equation under the nonzero continuous wave background. The Hirota-Satsuma equation relates to the Boussinesq equation through the Miura transformation. The bilinear form of Hirota-Satsuma equation is given through four identities under the bi-logarithmic transformation. N-soliton solutions and the first two soliton solutions are presented explicitly. The one-soliton solution admits dark or antidark soliton depending on the parametric condition, which decides the soliton type. The interactions between antidark and dark solitons, two dark solitons, two antidark solitons are discussed. It is found that the soliton solutions admit the algebraic solitons by taking the limit of wave numbers and proper phase constants. Meanwhile, the N-soliton solutions admit the localized wave in the form of breather and its interaction with the dark or antidark soliton. The corresponding excitation conditions of interaction solutions are presented. It is found that the interactions between solitons and breathers are elastic, which implies that the breathers and solitons can retain their identities after the interaction. The asymptotic analysis is implemented to verify the invariance of soliton characteristic.